scipy.stats.dirichlet#

scipy.stats.dirichlet = <scipy.stats._multivariate.dirichlet_gen object>[source]#

A Dirichlet random variable.

The alpha keyword specifies the concentration parameters of the distribution.

Added in version 0.15.0.

Parameters:
alphaarray_like

The concentration parameters. The number of entries determines the dimensionality of the distribution.

seed{None, int, np.random.RandomState, np.random.Generator}, optional

Used for drawing random variates. If seed is None, the RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a RandomState or Generator instance, then that object is used. Default is None.

Methods

pdf(x, alpha)

Probability density function.

logpdf(x, alpha)

Log of the probability density function.

rvs(alpha, size=1, random_state=None)

Draw random samples from a Dirichlet distribution.

mean(alpha)

The mean of the Dirichlet distribution

var(alpha)

The variance of the Dirichlet distribution

cov(alpha)

The covariance of the Dirichlet distribution

entropy(alpha)

Compute the differential entropy of the Dirichlet distribution.

Notes

Each \(\alpha\) entry must be positive. The distribution has only support on the simplex defined by

\[\sum_{i=1}^{K} x_i = 1\]

where \(0 < x_i < 1\).

If the quantiles don’t lie within the simplex, a ValueError is raised.

The probability density function for dirichlet is

\[f(x) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}\]

where

\[\mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)} {\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)}\]

and \(\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)\), the concentration parameters and \(K\) is the dimension of the space where \(x\) takes values.

Note that the dirichlet interface is somewhat inconsistent. The array returned by the rvs function is transposed with respect to the format expected by the pdf and logpdf.

Examples

>>> import numpy as np
>>> from scipy.stats import dirichlet

Generate a dirichlet random variable

>>> quantiles = np.array([0.2, 0.2, 0.6])  # specify quantiles
>>> alpha = np.array([0.4, 5, 15])  # specify concentration parameters
>>> dirichlet.pdf(quantiles, alpha)
0.2843831684937255

The same PDF but following a log scale

>>> dirichlet.logpdf(quantiles, alpha)
-1.2574327653159187

Once we specify the dirichlet distribution we can then calculate quantities of interest

>>> dirichlet.mean(alpha)  # get the mean of the distribution
array([0.01960784, 0.24509804, 0.73529412])
>>> dirichlet.var(alpha) # get variance
array([0.00089829, 0.00864603, 0.00909517])
>>> dirichlet.entropy(alpha)  # calculate the differential entropy
-4.3280162474082715

We can also return random samples from the distribution

>>> dirichlet.rvs(alpha, size=1, random_state=1)
array([[0.00766178, 0.24670518, 0.74563305]])
>>> dirichlet.rvs(alpha, size=2, random_state=2)
array([[0.01639427, 0.1292273 , 0.85437844],
       [0.00156917, 0.19033695, 0.80809388]])

Alternatively, the object may be called (as a function) to fix concentration parameters, returning a “frozen” Dirichlet random variable:

>>> rv = dirichlet(alpha)
>>> # Frozen object with the same methods but holding the given
>>> # concentration parameters fixed.
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